Carousels, Zindler curves and the floating body problem

Abstract

A carousel is a dynamical system that describes the movement of an equilateral linkage in which the midpoint of each rod travels parallel to it. They are closely related to the floating body problem. We prove, using the work of Auerbach, that any figure that floats in equilibrium in every position is drawn by a carousel. Of special interest are such figures with rational perimetral density of the floating chords, which are then drawn by carousels. In particular, we prove that for some perimetral densities the only such figure is the circle, as the problem suggests.

Self-Bäcklund curves in centroaffine geometry and Lamé's equation

“…Sections 3.4 and 3.5 concern centroaffine carrousels, self-Bäcklund curves with a rational rotation number (we call them carrousels because this term was used in [14,15] in the study of a similar problem in Euclidean geometry). We describe centroaffine carrousels as closed trajectories of a certain Hamiltonian vector field on the space of centroaffine 2n-gons.…”

“…We provide details in the first non-trivial case, n = 5. A similar approach to bicycle curves was developed in [14,15]. We also study the centroaffine curves that are c-related to central ellipses.…”

Spherical floating bodies

“…One of such results [6,Theortem 4] says that if a centrally symmetric body of revolution with δ = 1 2 floats indifferently stable in every direction, then it is a sphere. Another one in [2,Theorem 5] states that in dimension 2 the only figure that floats in equilibrium in every position and has perimetral density 3 1 3 or 1 4 is the circle (a more general result in this style can be found in [14]).…”

“…1 Some restriction on the body is requested to avoid trivial counterexamples. 2 In dimension 2 for δ = 1 2 given by [1,14] and for δ ∈ (0, 1 2 ) by [18,19]. In dimension 3 for δ ∈ (0, 1 2 ] by [21].…”

Remarks on “Floating Bodies in Neutral Equilibrium”

“…The problem as formulated is open to some interpretation. Responses containing counterexamples that appeared in [4] and in ensuing literature [5,6,7] were based on hydrostatic theory going back to Archimedes that took no account of surface tension. Independently in 2007 Mattie Sloss and I came on to the problem from a different point of view taking a partial account of surface tension, and published in [3] an affirmative answer to the question in that context, for smooth convex three-dimensional bodies.…”